Charlotte and Richmond are about 248 miles apart by road. Split the difference and you arrive near Raleigh, the city closest to the halfway point between them. That splits the trip almost evenly — about 3 hr 15 min from Charlotte and 3 hr 28 min from Richmond.
At this range, Raleigh works for a same-day meetup: close enough from both Charlotte and Richmond to meet for lunch or an afternoon and still be home by evening.
If Raleigh doesn't have what you're after, Washington and Pittsburgh are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Charlotte and Richmond.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.